TESTING/EIG/sget52.f(3)

NAME
TESTING/EIG/sget52.f

SYNOPSIS

  Functions/Subroutines
subroutine sget52 (left, n, a, lda, b, ldb, e, lde, alphar, alphai, beta, work, result)
SGET52

Function/Subroutine Documentation

  subroutine sget52 (logical left, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( lde, * ) e, integer lde, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( * ) work, real, dimension( 2 ) result)

SGET52

Purpose:

    !>
    !> SGET52  does an eigenvector check for the generalized eigenvalue
    !> problem.
    !>
    !> The basic test for right eigenvectors is:
    !>
    !>                           | b(j) A E(j) -  a(j) B E(j) |
    !>         RESULT(1) = max   -------------------------------
    !>                      j    n ulp max( |b(j) A|, |a(j) B| )
    !>
    !> using the 1-norm.  Here, a(j)/b(j) = w is the j-th generalized
    !> eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
    !> generalized eigenvalue of m A - B.
    !>
    !> For real eigenvalues, the test is straightforward.  For complex
    !> eigenvalues, E(j) and a(j) are complex, represented by
    !> Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
    !> eigenvector becomes
    !>
    !>                 max( |Wr|, |Wi| )
    !>     --------------------------------------------
    !>     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
    !>
    !> where
    !>
    !>     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
    !>
    !>     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
    !>
    !>                         T   T  _
    !> For left eigenvectors, A , B , a, and b  are used.
    !>
    !> SGET52 also tests the normalization of E.  Each eigenvector is
    !> supposed to be normalized so that the maximum
    !> of its elements is 1, where in this case,
    !> of a complex value x is  |Re(x)| + |Im(x)| ; let us call this
    !> maximum  norm of a vector v  M(v).
    !> if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
    !> vector.  The normalization test is:
    !>
    !>         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )
    !>                    eigenvectors v(j)
    !>

Parameters
    LEFT

    !>          LEFT is LOGICAL
    !>          =.TRUE.:  The eigenvectors in the columns of E are assumed
    !>                    to be *left* eigenvectors.
    !>          =.FALSE.: The eigenvectors in the columns of E are assumed
    !>                    to be *right* eigenvectors.
    !>

    N

    !>          N is INTEGER
    !>          The size of the matrices.  If it is zero, SGET52 does
    !>          nothing.  It must be at least zero.
    !>

    A

    !>          A is REAL array, dimension (LDA, N)
    !>          The matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of A.  It must be at least 1
    !>          and at least N.
    !>

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          The matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of B.  It must be at least 1
    !>          and at least N.
    !>

    E

    !>          E is REAL array, dimension (LDE, N)
    !>          The matrix of eigenvectors.  It must be O( 1 ).  Complex
    !>          eigenvalues and eigenvectors always come in pairs, the
    !>          eigenvalue and its conjugate being stored in adjacent
    !>          elements of ALPHAR, ALPHAI, and BETA.  Thus, if a(j)/b(j)
    !>          and a(j+1)/b(j+1) are a complex conjugate pair of
    !>          generalized eigenvalues, then E(,j) contains the real part
    !>          of the eigenvector and E(,j+1) contains the imaginary part.
    !>          Note that whether E(,j) is a real eigenvector or part of a
    !>          complex one is specified by whether ALPHAI(j) is zero or not.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of E.  It must be at least 1 and at
    !>          least N.
    !>

    ALPHAR

    !>          ALPHAR is REAL array, dimension (N)
    !>          The real parts of the values a(j) as described above, which,
    !>          along with b(j), define the generalized eigenvalues.
    !>          Complex eigenvalues always come in complex conjugate pairs
    !>          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
    !>          elements in ALPHAR, ALPHAI, and BETA.  Thus, if the j-th
    !>          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
    !>          is assumed to be equal to ALPHAR(j)/BETA(j).
    !>

    ALPHAI

    !>          ALPHAI is REAL array, dimension (N)
    !>          The imaginary parts of the values a(j) as described above,
    !>          which, along with b(j), define the generalized eigenvalues.
    !>          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
    !>          is part of a complex conjugate pair.  Complex eigenvalues
    !>          always come in complex conjugate pairs a(j)/b(j) and
    !>          a(j+1)/b(j+1), which are stored in adjacent elements in
    !>          ALPHAR, ALPHAI, and BETA.  Thus, if the j-th and (j+1)-st
    !>          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
    !>          be equal to  -ALPHAI(j)/BETA(j).  Also, nonzero values in
    !>          ALPHAI are assumed to always come in adjacent pairs.
    !>

    BETA

    !>          BETA is REAL array, dimension (N)
    !>          The values b(j) as described above, which, along with a(j),
    !>          define the generalized eigenvalues.
    !>

    WORK

    !>          WORK is REAL array, dimension (N**2+N)
    !>

    RESULT

    !>          RESULT is REAL array, dimension (2)
    !>          The values computed by the test described above.  If A E or
    !>          B E is likely to overflow, then RESULT(1:2) is set to
    !>          10 / ulp.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 197 of file sget52.f.

Author

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